Consider the mappings I : P4 P5 and D : P P5 defined by a1 a3 4 0212 +2 +2 + 3 4 I(p(x)) = ax + a4 5 X 5 D(p(x)) = a +2ax +3a3x +4a4x. where p(x) = a + ax + ax + a3x + ax. You may assume that both these mappings are linear. Let T : P4 P5 be defined by T(p(x)) = I(p(x)) 3xD(p(x)). - (a) Prove that T is linear using the definition of linearity. Hint: You may use the fact that D and I are linear. (b) Determine a basis for the kernel of T and a basis for the range of T. Note: Make sure that you don't just find a spanning set, but you also prove it is linearly independent.